Simplify; express your answer in exponential form. Assume $q\neq 0, p\neq 0$. $\dfrac{{(q^{-4}p^{4})^{5}}}{{q^{-1}p^{-3}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{-4}p^{4})^{5} = (q^{-4})^{5}(p^{4})^{5}}$ On the left, we have ${q^{-4}}$ to the exponent ${5}$ . Now ${-4 \times 5 = -20}$ , so ${(q^{-4})^{5} = q^{-20}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{-4}p^{4})^{5}}}{{q^{-1}p^{-3}}} = \dfrac{{q^{-20}p^{20}}}{{q^{-1}p^{-3}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{-20}p^{20}}}{{q^{-1}p^{-3}}} = \dfrac{{q^{-20}}}{{q^{-1}}} \cdot \dfrac{{p^{20}}}{{p^{-3}}} = q^{{-20} - {(-1)}} \cdot p^{{20} - {(-3)}} = q^{-19}p^{23}$